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On the Conserved Caginalp Phase-Field System with Logarithmic Potentials Based on the Maxwell–Cattaneo Law with Two Temperatures

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Abstract

Our aim in this article is to study generalizations of the conserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures for heat conduction and with logarithmic nonlinear terms. We obtain well-posedness results and study the asymptotic behavior of the associated system. In particular, we prove the existence of the global attractor and prove the strict separation to the pure phases in two space dimensions. Furthermore, we give some numerical simulations, obtained with the FreeFem++ software [23], comparing the conserved Caginalp phase-field type model with regular and with logarithmic nonlinear terms.

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Acknowledgements

The authors wish to thank the referees for their careful reading of the manuscript and many useful comments which helped improve it.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan, China

    Alain Miranville

  2. Laboratoire de Mathématiques et Applications, Université de Poitiers, UMR CNRS 7348 - SP2MI Boulevard Marie et Pierre Curie - Téléport 2, 86962, Chasseneuil Futuroscope Cedex, France

    Ahmad Makki & Alain Miranville

  3. Fudan University (Fudan Fellow), Shanghai, China

    Alain Miranville

  4. School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, China

    Alain Miranville

  5. Laboratoire de Mathématiques Raphaël Salem, Université de Rouen Normandie, CNRS UMR 6085, Avenue de l’Université, BP 12, 76801, Saint-Étienne-du-Rouvray, France

    Georges Sadaka

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  1. Ahmad Makki
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Correspondence to Ahmad Makki.

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Makki, A., Miranville, A. & Sadaka, G. On the Conserved Caginalp Phase-Field System with Logarithmic Potentials Based on the Maxwell–Cattaneo Law with Two Temperatures. Appl Math Optim 84, 1285–1316 (2021). https://doi.org/10.1007/s00245-020-09677-0

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